Question: Find the slope and y-intercept of the line that is ${\text{parallel}}$ to $\enspace {y = \dfrac{1}{3}x + 1}\enspace$ and passes through the point ${(-5, -7)}$. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$
Solution: Parallel lines have the same slope. The slope of the blue line is ${\dfrac{1}{3}}$ , so the equation of our parallel line will be of the form $\enspace {y = \dfrac{1}{3}x + b}\enspace$ We can plug our point, $(-5, -7)$ , into this equation to solve for ${b}$ , the y-intercept. $-7 = {\dfrac{1}{3}}(-5) + {b}$ $-7 = -\dfrac{5}{3} + {b}$ $-7 + \dfrac{5}{3} = {b} = -\dfrac{16}{3}$ The equation of the parallel line is $\enspace {y = \dfrac{1}{3}x - \dfrac{16}{3}}\enspace$. ${m = \dfrac{1}{3}, \enspace b = -\dfrac{16}{3}}$